Light from a sodium lamp ( wavelength= 589 nm) illuminates a narrow slit and is observed on a screen 75 cm behind the slit. The dstance between the first and third dark fringes is 7.5 mm. What is the width (in mm) of the slit?
Cant seem to get the answer =/
To find the width of the slit, we can use the concept of interference and diffraction.
Given:
Wavelength of light, λ = 589 nm = 589 × 10^(-9) m
Distance between the first and third dark fringes, D = 7.5 mm = 7.5 × 10^(-3) m
Distance between the slit and the screen, L = 75 cm = 75 × 10^(-2) m
We can use the formula for the fringe separation in a diffraction pattern:
d sinθ = mλ
Where:
d is the width of the slit
θ is the angle between the central maximum and the mth dark fringe
m is the order of the fringe
Since we are given the distance between the first and third dark fringes, we can say that m = 3 - 1 = 2 (since we are counting the dark fringes from the central maximum).
Now, let's rearrange the formula to solve for d:
d = (mλ) / sinθ
To find θ, we can use the small angle approximation which states that for small angles sinθ ≈ tanθ ≈ θ.
Therefore,
d ≈ (mλ) / θ
Now, we need to find the angle θ. We can use the formula for finding the angle in a triangle:
tanθ = opposite / adjacent
In this case, the opposite side is D and the adjacent side is L. Therefore,
tanθ = D / L
Solving for θ:
θ = tan^(-1) (D / L)
Now, we can substitute the given values into the formula to find θ:
θ = tan^(-1) [(7.5 × 10^(-3)) / (75 × 10^(-2))]
θ ≈ tan^(-1) (0.01)
θ ≈ 0.57296 radians
Now, substitute the values of θ, m, and λ into the formula for d:
d = (2 × 589 × 10^(-9)) / sin(0.57296)
d ≈ 2 × 589 × 10^(-9) / sin(0.57296)
d ≈ 2.056 × 10^(-3) m
Finally, convert the answer to millimeters:
d ≈ 2.056 × 10^(-3) m × 10^3 mm/m
d ≈ 2.056 mm
So, the width of the slit is approximately 2.056 mm.
To find the width of the slit, we can use the concept of diffraction and the equation for the angular position of the dark fringes.
The formula for the angular position of the dark fringes caused by a single slit is given by:
sin(θ) = (m * λ) / w
where:
θ is the angular position of the dark fringe,
m is the order of the dark fringe (here, m = 1 since we are considering the first dark fringe),
λ is the wavelength of light (589 nm in this case), and
w is the width of the slit (which we need to find).
First, let's convert the wavelength from nanometers to meters:
λ = 589 nm = 589 × 10^(-9) m
Now, let's rearrange the equation to solve for the width of the slit (w):
w = (m * λ) / sin(θ).
We need the value of θ to calculate the width of the slit. We can find θ using the distance between the first and third dark fringes, as given in the problem.
The formula to calculate the distance between the dark fringes is given by:
y = ((m * λ) * L) / w
where:
y is the distance between the dark fringes (7.5 mm in this case),
m is the order of the dark fringe (here, m = 3 since we are considering the third dark fringe),
L is the distance between the slit and the screen (75 cm = 750 mm in this case), and
w is the width of the slit.
Rearranging the equation, we can solve for the angular position of the dark fringe (θ):
sin(θ) = (m * λ) / (y * L / w)
Now we can substitute the values into the equation and solve for θ:
sin(θ) = (3 * 589 × 10^(-9)) / (7.5 × 750 / w)
sin(θ) = (3 * 589 × 10^(-9) * w) / (7.5 × 750)
we know that sin(θ) can be approximated as θ for small angles.
θ = (3 * 589 × 10^(-9) * w) / (7.5 × 750)
Now we have θ, we can use this value to calculate w.
w = (m * λ) / sin(θ)
w = (1 * 589 × 10^(-9)) / sin(θ)
Note: The measurement units may vary, but make sure to use consistent units for accurate results.
I hope this explanation helps you understand how to solve the problem. If you have any further questions, feel free to ask!
Read the "single slit diffraction" section of
http://en.wikipedia.org/wiki/Diffraction
The formula you need can be found there.
The angle theta at any distance x from the central maximum is
theta = tan^-1 (x/75)
where x is in cm